Every time I try to understand the 4th+ dimension, my brain just completely breaks and I'm left feeling angry. I'm a highly visual thinker so it's difficult for me to grasp concepts like that.
Someone once told me that in the same way a 3D object casts a 2D shadow, a 4D object casts a 3D shadow. I just...can't. I can't wrap my head around that no matter how hard I try.
Here is a much less tricky version: A car’s dimensions can include its length, height, and width, but also the amount of gas in the tank (4th), the position of the throttle pedal (5th) and brake pedal (6th), its weight (7th), the number of passengers (8th), etc etc.
I don’t think that works because the N-1 dimension is contained in the N dimension. But all those things described are disjoint. Now you might be able to say, “the mass of all objects in the universe” but is it a dimension? I dunno I’ll leave that to the topologists!
Edit: I think you have to have the similar type of units to increase the dimensions. Like the article talks about N dimensional space (e.g. point, line, plane, etc). To consider what mass of an object would be you’d have 1 dimensional mass; 2D mass; …; ND mass whatever that would be.
> I don’t think that works because the N-1 dimension is contained in the N dimension.
I'm not sure that's right. Each dimension is linearly independent [0] from the others, which means e.g. you can't add up a bunch of width and get height, or add up a bunch of length and get width. So in an important sense, they're not contained within each other.
You might be thinking of how a 2D plane contains the first dimension within it, but that's not the 2nd dimension... that's two-dimensional (a combination of two dimensions).
Hmm yeah that what I’m confusing. I suppose it’s misleading when folks say The Fourth Dimension since it depends on context! But in the case of the article the question becomes how do we visualize four or more dimensions? What the dimensions represent doesn’t necessarily matter unless there’s a specific problem being solved.
The car example works as a vector space, but not an inner-product space. There's no sensible way to rotate the length of the car to gasoline in the tank
That's right! The less pithy answer: objects are the things that stay invariant under a transformation. In that sense then car length and gas are somehow MORE than linearly independent - they are "geometrically independent".
(Note there is a relationship between the length of the car and the gas tank when the car is moving fast (close to c) in your reference frame - but AFAIK there aren't any useful geometric transformations that depend on that fact. :)
Isn't that the definition of rotation, the "exchanging one dimension for another"? In 2-D it keeps a point invariant, in 3-D a line, and in N-D it keeps the N-1 object invariant. (Side note: the more basic operation is reflection, since you can get rotation from two reflections)
You can't rotate around an arbitrary vector and have current brake pedal position determine the number of passengers.
There's a fundamental difference between a "set of independent variables" and a physical space with geometric transformations. The latter is far more complex to visualise.
I had a notion for a physical theory of everything, but I got hopelessly lost once I wandered into 6-dimensional knot theory.
You may try to imagine 3d objects (shadows of 4d) in consecutive slices. E.g. the foreground 3d scene contains a dot, middle scenes contain bigger and then again smaller spheres down to the background scenewith a dot. By looking at the entire setup from the side you’ll notice that spheres grow and shrink in the same fashion the 2d sphere’s x/y’s do. Similar to this, but in 3d: https://www.google.ru/search?q=sphere+slices&tbm=isch
A 4-cube would look like just few identical 3-cubes from the side, if oriented face-to-3-space. If not, it will be series of prisms or platonic solid alikes.
It doesn’t help much with shapes that have complex 4d-edges, or with higher dimensions, but may give an idea. Also, open two gifs from this link http://www.math.union.edu/~dpvc/math/4D/folding/welcome.html side by side and you’ll quickly get the concept of rotation of a solid around 4th axis.
With the 4th dimension, you can cheat a bit and use time itself. For instance, a 4-D sphere is something that starts at a point, grows to a sphere, then shrinks again.
It's not perfect. We can rotate in the normal 3D space but you don't get proper 4D rotation with this technique. Nor does this give you a good intuition about that shadow.
But it's something.
Gives you basically nothing when it comes to the 5th dimension, though, because I'm fresh out of spacetime at that point.
I seriously question that. Can you point us to any evidence that people of average IQ are most often visual thinkers? (Or most often say that they're visual thinkers, if that's where you were going with this.)
I cannot. I am merely calling out that fact that "I'm a visual thinker" is almost always used as a justification for not being able to keep up. Sort of like saying "My other car is a Porsche."
Is it not possible that different people will have an easier or harder time understanding different kinds of concepts? Maybe IQ favors a particular type of "non-visual" thinking. Or maybe IQ is just another pile of bullshit from the field of physchology.
Someone once told me that in the same way a 3D object casts a 2D shadow, a 4D object casts a 3D shadow. I just...can't. I can't wrap my head around that no matter how hard I try.